Properties

Label 7800.2.a.bt.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.30056.1
Defining polynomial: \(x^{4} - 11 x^{2} + 26\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.85431\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +5.51003 q^{11} +1.00000 q^{13} -6.21728 q^{17} +3.09417 q^{21} +4.21728 q^{23} -1.00000 q^{27} -1.70861 q^{29} -5.51003 q^{33} -4.02893 q^{37} -1.00000 q^{39} +0.198578 q^{41} -5.70861 q^{43} -6.41586 q^{47} +2.57391 q^{49} +6.21728 q^{51} +4.02893 q^{53} -7.53896 q^{59} +11.9259 q^{61} -3.09417 q^{63} +4.83172 q^{67} -4.21728 q^{69} +7.98977 q^{71} +9.31145 q^{73} -17.0490 q^{77} +8.51139 q^{79} +1.00000 q^{81} -14.7926 q^{83} +1.70861 q^{87} -9.40699 q^{89} -3.09417 q^{91} +14.3026 q^{97} +5.51003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 7q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 7q^{7} + 4q^{9} + q^{11} + 4q^{13} - 3q^{17} + 7q^{21} - 5q^{23} - 4q^{27} + 8q^{29} - q^{33} - 5q^{37} - 4q^{39} + 7q^{41} - 8q^{43} - 10q^{47} + 9q^{49} + 3q^{51} + 5q^{53} + 2q^{59} + 11q^{61} - 7q^{63} - 12q^{67} + 5q^{69} + 15q^{71} + 10q^{73} - 15q^{77} - q^{79} + 4q^{81} - 22q^{83} - 8q^{87} + 9q^{89} - 7q^{91} - q^{97} + q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.09417 −1.16949 −0.584744 0.811218i \(-0.698805\pi\)
−0.584744 + 0.811218i \(0.698805\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.51003 1.66134 0.830669 0.556767i \(-0.187958\pi\)
0.830669 + 0.556767i \(0.187958\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.21728 −1.50791 −0.753956 0.656925i \(-0.771857\pi\)
−0.753956 + 0.656925i \(0.771857\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.09417 0.675204
\(22\) 0 0
\(23\) 4.21728 0.879364 0.439682 0.898154i \(-0.355091\pi\)
0.439682 + 0.898154i \(0.355091\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.70861 −0.317281 −0.158640 0.987336i \(-0.550711\pi\)
−0.158640 + 0.987336i \(0.550711\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −5.51003 −0.959173
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.02893 −0.662352 −0.331176 0.943569i \(-0.607445\pi\)
−0.331176 + 0.943569i \(0.607445\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0.198578 0.0310127 0.0155064 0.999880i \(-0.495064\pi\)
0.0155064 + 0.999880i \(0.495064\pi\)
\(42\) 0 0
\(43\) −5.70861 −0.870555 −0.435277 0.900296i \(-0.643350\pi\)
−0.435277 + 0.900296i \(0.643350\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.41586 −0.935849 −0.467925 0.883768i \(-0.654998\pi\)
−0.467925 + 0.883768i \(0.654998\pi\)
\(48\) 0 0
\(49\) 2.57391 0.367702
\(50\) 0 0
\(51\) 6.21728 0.870593
\(52\) 0 0
\(53\) 4.02893 0.553416 0.276708 0.960954i \(-0.410756\pi\)
0.276708 + 0.960954i \(0.410756\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.53896 −0.981489 −0.490745 0.871303i \(-0.663275\pi\)
−0.490745 + 0.871303i \(0.663275\pi\)
\(60\) 0 0
\(61\) 11.9259 1.52695 0.763477 0.645835i \(-0.223491\pi\)
0.763477 + 0.645835i \(0.223491\pi\)
\(62\) 0 0
\(63\) −3.09417 −0.389829
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.83172 0.590288 0.295144 0.955453i \(-0.404632\pi\)
0.295144 + 0.955453i \(0.404632\pi\)
\(68\) 0 0
\(69\) −4.21728 −0.507701
\(70\) 0 0
\(71\) 7.98977 0.948211 0.474106 0.880468i \(-0.342772\pi\)
0.474106 + 0.880468i \(0.342772\pi\)
\(72\) 0 0
\(73\) 9.31145 1.08982 0.544912 0.838494i \(-0.316563\pi\)
0.544912 + 0.838494i \(0.316563\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −17.0490 −1.94291
\(78\) 0 0
\(79\) 8.51139 0.957607 0.478803 0.877922i \(-0.341071\pi\)
0.478803 + 0.877922i \(0.341071\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.7926 −1.62369 −0.811847 0.583871i \(-0.801538\pi\)
−0.811847 + 0.583871i \(0.801538\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.70861 0.183182
\(88\) 0 0
\(89\) −9.40699 −0.997139 −0.498569 0.866850i \(-0.666141\pi\)
−0.498569 + 0.866850i \(0.666141\pi\)
\(90\) 0 0
\(91\) −3.09417 −0.324358
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.3026 1.45221 0.726104 0.687585i \(-0.241329\pi\)
0.726104 + 0.687585i \(0.241329\pi\)
\(98\) 0 0
\(99\) 5.51003 0.553779
\(100\) 0 0
\(101\) −6.72595 −0.669257 −0.334628 0.942350i \(-0.608611\pi\)
−0.334628 + 0.942350i \(0.608611\pi\)
\(102\) 0 0
\(103\) −17.0779 −1.68274 −0.841369 0.540461i \(-0.818250\pi\)
−0.841369 + 0.540461i \(0.818250\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.63450 0.738055 0.369027 0.929418i \(-0.379691\pi\)
0.369027 + 0.929418i \(0.379691\pi\)
\(108\) 0 0
\(109\) −5.02006 −0.480835 −0.240417 0.970670i \(-0.577284\pi\)
−0.240417 + 0.970670i \(0.577284\pi\)
\(110\) 0 0
\(111\) 4.02893 0.382409
\(112\) 0 0
\(113\) 16.6229 1.56375 0.781876 0.623434i \(-0.214263\pi\)
0.781876 + 0.623434i \(0.214263\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 19.2373 1.76348
\(120\) 0 0
\(121\) 19.3604 1.76004
\(122\) 0 0
\(123\) −0.198578 −0.0179052
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.87689 −0.255283 −0.127642 0.991820i \(-0.540741\pi\)
−0.127642 + 0.991820i \(0.540741\pi\)
\(128\) 0 0
\(129\) 5.70861 0.502615
\(130\) 0 0
\(131\) −16.5199 −1.44335 −0.721674 0.692233i \(-0.756627\pi\)
−0.721674 + 0.692233i \(0.756627\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.8504 1.09789 0.548943 0.835860i \(-0.315031\pi\)
0.548943 + 0.835860i \(0.315031\pi\)
\(138\) 0 0
\(139\) −17.0490 −1.44608 −0.723038 0.690808i \(-0.757255\pi\)
−0.723038 + 0.690808i \(0.757255\pi\)
\(140\) 0 0
\(141\) 6.41586 0.540313
\(142\) 0 0
\(143\) 5.51003 0.460772
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.57391 −0.212293
\(148\) 0 0
\(149\) 19.5679 1.60306 0.801532 0.597952i \(-0.204019\pi\)
0.801532 + 0.597952i \(0.204019\pi\)
\(150\) 0 0
\(151\) −9.22887 −0.751035 −0.375518 0.926815i \(-0.622535\pi\)
−0.375518 + 0.926815i \(0.622535\pi\)
\(152\) 0 0
\(153\) −6.21728 −0.502637
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.89423 −0.310794 −0.155397 0.987852i \(-0.549666\pi\)
−0.155397 + 0.987852i \(0.549666\pi\)
\(158\) 0 0
\(159\) −4.02893 −0.319515
\(160\) 0 0
\(161\) −13.0490 −1.02840
\(162\) 0 0
\(163\) −0.237741 −0.0186213 −0.00931067 0.999957i \(-0.502964\pi\)
−0.00931067 + 0.999957i \(0.502964\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.8304 −0.915460 −0.457730 0.889091i \(-0.651337\pi\)
−0.457730 + 0.889091i \(0.651337\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.6634 0.886754 0.443377 0.896335i \(-0.353780\pi\)
0.443377 + 0.896335i \(0.353780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.53896 0.566663
\(178\) 0 0
\(179\) −12.5403 −0.937308 −0.468654 0.883382i \(-0.655261\pi\)
−0.468654 + 0.883382i \(0.655261\pi\)
\(180\) 0 0
\(181\) −5.44343 −0.404607 −0.202303 0.979323i \(-0.564843\pi\)
−0.202303 + 0.979323i \(0.564843\pi\)
\(182\) 0 0
\(183\) −11.9259 −0.881587
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −34.2574 −2.50515
\(188\) 0 0
\(189\) 3.09417 0.225068
\(190\) 0 0
\(191\) 5.22887 0.378348 0.189174 0.981944i \(-0.439419\pi\)
0.189174 + 0.981944i \(0.439419\pi\)
\(192\) 0 0
\(193\) 17.5287 1.26175 0.630873 0.775886i \(-0.282697\pi\)
0.630873 + 0.775886i \(0.282697\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.4764 −1.60138 −0.800690 0.599079i \(-0.795534\pi\)
−0.800690 + 0.599079i \(0.795534\pi\)
\(198\) 0 0
\(199\) −16.9143 −1.19902 −0.599511 0.800366i \(-0.704638\pi\)
−0.599511 + 0.800366i \(0.704638\pi\)
\(200\) 0 0
\(201\) −4.83172 −0.340803
\(202\) 0 0
\(203\) 5.28674 0.371056
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.21728 0.293121
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −11.8391 −0.815037 −0.407518 0.913197i \(-0.633606\pi\)
−0.407518 + 0.913197i \(0.633606\pi\)
\(212\) 0 0
\(213\) −7.98977 −0.547450
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9.31145 −0.629210
\(220\) 0 0
\(221\) −6.21728 −0.418219
\(222\) 0 0
\(223\) 4.48246 0.300168 0.150084 0.988673i \(-0.452046\pi\)
0.150084 + 0.988673i \(0.452046\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.35799 −0.421995 −0.210997 0.977487i \(-0.567671\pi\)
−0.210997 + 0.977487i \(0.567671\pi\)
\(228\) 0 0
\(229\) 6.45502 0.426560 0.213280 0.976991i \(-0.431585\pi\)
0.213280 + 0.976991i \(0.431585\pi\)
\(230\) 0 0
\(231\) 17.0490 1.12174
\(232\) 0 0
\(233\) −6.40290 −0.419468 −0.209734 0.977758i \(-0.567260\pi\)
−0.209734 + 0.977758i \(0.567260\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.51139 −0.552874
\(238\) 0 0
\(239\) −1.55793 −0.100774 −0.0503872 0.998730i \(-0.516046\pi\)
−0.0503872 + 0.998730i \(0.516046\pi\)
\(240\) 0 0
\(241\) −14.1911 −0.914127 −0.457064 0.889434i \(-0.651099\pi\)
−0.457064 + 0.889434i \(0.651099\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 14.7926 0.937440
\(250\) 0 0
\(251\) 22.7082 1.43333 0.716665 0.697418i \(-0.245668\pi\)
0.716665 + 0.697418i \(0.245668\pi\)
\(252\) 0 0
\(253\) 23.2373 1.46092
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.4172 −1.33597 −0.667985 0.744175i \(-0.732843\pi\)
−0.667985 + 0.744175i \(0.732843\pi\)
\(258\) 0 0
\(259\) 12.4662 0.774613
\(260\) 0 0
\(261\) −1.70861 −0.105760
\(262\) 0 0
\(263\) −26.4951 −1.63376 −0.816880 0.576807i \(-0.804298\pi\)
−0.816880 + 0.576807i \(0.804298\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.40699 0.575698
\(268\) 0 0
\(269\) 14.7259 0.897857 0.448928 0.893568i \(-0.351806\pi\)
0.448928 + 0.893568i \(0.351806\pi\)
\(270\) 0 0
\(271\) −25.8491 −1.57022 −0.785109 0.619357i \(-0.787393\pi\)
−0.785109 + 0.619357i \(0.787393\pi\)
\(272\) 0 0
\(273\) 3.09417 0.187268
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.1227 1.20906 0.604528 0.796584i \(-0.293362\pi\)
0.604528 + 0.796584i \(0.293362\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.3954 −0.978067 −0.489034 0.872265i \(-0.662650\pi\)
−0.489034 + 0.872265i \(0.662650\pi\)
\(282\) 0 0
\(283\) −14.5650 −0.865802 −0.432901 0.901441i \(-0.642510\pi\)
−0.432901 + 0.901441i \(0.642510\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.614436 −0.0362690
\(288\) 0 0
\(289\) 21.6546 1.27380
\(290\) 0 0
\(291\) −14.3026 −0.838432
\(292\) 0 0
\(293\) −26.6995 −1.55980 −0.779900 0.625904i \(-0.784730\pi\)
−0.779900 + 0.625904i \(0.784730\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.51003 −0.319724
\(298\) 0 0
\(299\) 4.21728 0.243892
\(300\) 0 0
\(301\) 17.6634 1.01810
\(302\) 0 0
\(303\) 6.72595 0.386396
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.4230 0.651943 0.325972 0.945380i \(-0.394309\pi\)
0.325972 + 0.945380i \(0.394309\pi\)
\(308\) 0 0
\(309\) 17.0779 0.971529
\(310\) 0 0
\(311\) 19.4546 1.10317 0.551585 0.834119i \(-0.314023\pi\)
0.551585 + 0.834119i \(0.314023\pi\)
\(312\) 0 0
\(313\) −32.5805 −1.84156 −0.920778 0.390087i \(-0.872445\pi\)
−0.920778 + 0.390087i \(0.872445\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −31.3781 −1.76237 −0.881184 0.472774i \(-0.843253\pi\)
−0.881184 + 0.472774i \(0.843253\pi\)
\(318\) 0 0
\(319\) −9.41450 −0.527111
\(320\) 0 0
\(321\) −7.63450 −0.426116
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.02006 0.277610
\(328\) 0 0
\(329\) 19.8518 1.09446
\(330\) 0 0
\(331\) −30.4951 −1.67616 −0.838082 0.545544i \(-0.816323\pi\)
−0.838082 + 0.545544i \(0.816323\pi\)
\(332\) 0 0
\(333\) −4.02893 −0.220784
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.8765 −0.864848 −0.432424 0.901670i \(-0.642342\pi\)
−0.432424 + 0.901670i \(0.642342\pi\)
\(338\) 0 0
\(339\) −16.6229 −0.902832
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.6951 0.739465
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.1069 −1.13308 −0.566538 0.824036i \(-0.691717\pi\)
−0.566538 + 0.824036i \(0.691717\pi\)
\(348\) 0 0
\(349\) 4.82899 0.258490 0.129245 0.991613i \(-0.458745\pi\)
0.129245 + 0.991613i \(0.458745\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −19.4359 −1.03447 −0.517235 0.855844i \(-0.673039\pi\)
−0.517235 + 0.855844i \(0.673039\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −19.2373 −1.01815
\(358\) 0 0
\(359\) −27.2271 −1.43699 −0.718496 0.695531i \(-0.755169\pi\)
−0.718496 + 0.695531i \(0.755169\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −19.3604 −1.01616
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.7008 0.819577 0.409788 0.912181i \(-0.365603\pi\)
0.409788 + 0.912181i \(0.365603\pi\)
\(368\) 0 0
\(369\) 0.198578 0.0103376
\(370\) 0 0
\(371\) −12.4662 −0.647214
\(372\) 0 0
\(373\) −3.86952 −0.200356 −0.100178 0.994970i \(-0.531941\pi\)
−0.100178 + 0.994970i \(0.531941\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.70861 −0.0879979
\(378\) 0 0
\(379\) −7.30149 −0.375052 −0.187526 0.982260i \(-0.560047\pi\)
−0.187526 + 0.982260i \(0.560047\pi\)
\(380\) 0 0
\(381\) 2.87689 0.147388
\(382\) 0 0
\(383\) 37.2094 1.90131 0.950655 0.310250i \(-0.100413\pi\)
0.950655 + 0.310250i \(0.100413\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.70861 −0.290185
\(388\) 0 0
\(389\) 30.1227 1.52728 0.763641 0.645641i \(-0.223410\pi\)
0.763641 + 0.645641i \(0.223410\pi\)
\(390\) 0 0
\(391\) −26.2200 −1.32600
\(392\) 0 0
\(393\) 16.5199 0.833317
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.25780 0.0631274 0.0315637 0.999502i \(-0.489951\pi\)
0.0315637 + 0.999502i \(0.489951\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.6443 1.03093 0.515464 0.856911i \(-0.327619\pi\)
0.515464 + 0.856911i \(0.327619\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −22.1995 −1.10039
\(408\) 0 0
\(409\) −31.3820 −1.55174 −0.775870 0.630893i \(-0.782689\pi\)
−0.775870 + 0.630893i \(0.782689\pi\)
\(410\) 0 0
\(411\) −12.8504 −0.633864
\(412\) 0 0
\(413\) 23.3269 1.14784
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 17.0490 0.834893
\(418\) 0 0
\(419\) −19.5549 −0.955321 −0.477661 0.878544i \(-0.658515\pi\)
−0.477661 + 0.878544i \(0.658515\pi\)
\(420\) 0 0
\(421\) −5.07793 −0.247483 −0.123742 0.992314i \(-0.539489\pi\)
−0.123742 + 0.992314i \(0.539489\pi\)
\(422\) 0 0
\(423\) −6.41586 −0.311950
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −36.9008 −1.78575
\(428\) 0 0
\(429\) −5.51003 −0.266027
\(430\) 0 0
\(431\) 7.52900 0.362659 0.181330 0.983422i \(-0.441960\pi\)
0.181330 + 0.983422i \(0.441960\pi\)
\(432\) 0 0
\(433\) −37.4821 −1.80127 −0.900637 0.434573i \(-0.856899\pi\)
−0.900637 + 0.434573i \(0.856899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 13.4836 0.643535 0.321767 0.946819i \(-0.395723\pi\)
0.321767 + 0.946819i \(0.395723\pi\)
\(440\) 0 0
\(441\) 2.57391 0.122567
\(442\) 0 0
\(443\) −6.61171 −0.314132 −0.157066 0.987588i \(-0.550204\pi\)
−0.157066 + 0.987588i \(0.550204\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −19.5679 −0.925530
\(448\) 0 0
\(449\) 5.24183 0.247377 0.123689 0.992321i \(-0.460528\pi\)
0.123689 + 0.992321i \(0.460528\pi\)
\(450\) 0 0
\(451\) 1.09417 0.0515226
\(452\) 0 0
\(453\) 9.22887 0.433610
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.69702 −0.406829 −0.203415 0.979093i \(-0.565204\pi\)
−0.203415 + 0.979093i \(0.565204\pi\)
\(458\) 0 0
\(459\) 6.21728 0.290198
\(460\) 0 0
\(461\) −3.11287 −0.144981 −0.0724905 0.997369i \(-0.523095\pi\)
−0.0724905 + 0.997369i \(0.523095\pi\)
\(462\) 0 0
\(463\) −33.8328 −1.57234 −0.786172 0.618008i \(-0.787940\pi\)
−0.786172 + 0.618008i \(0.787940\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0112 1.29620 0.648102 0.761554i \(-0.275563\pi\)
0.648102 + 0.761554i \(0.275563\pi\)
\(468\) 0 0
\(469\) −14.9502 −0.690335
\(470\) 0 0
\(471\) 3.89423 0.179437
\(472\) 0 0
\(473\) −31.4546 −1.44629
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.02893 0.184472
\(478\) 0 0
\(479\) 0.424328 0.0193880 0.00969401 0.999953i \(-0.496914\pi\)
0.00969401 + 0.999953i \(0.496914\pi\)
\(480\) 0 0
\(481\) −4.02893 −0.183703
\(482\) 0 0
\(483\) 13.0490 0.593750
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −32.5720 −1.47598 −0.737989 0.674813i \(-0.764224\pi\)
−0.737989 + 0.674813i \(0.764224\pi\)
\(488\) 0 0
\(489\) 0.237741 0.0107510
\(490\) 0 0
\(491\) −1.36659 −0.0616734 −0.0308367 0.999524i \(-0.509817\pi\)
−0.0308367 + 0.999524i \(0.509817\pi\)
\(492\) 0 0
\(493\) 10.6229 0.478432
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.7217 −1.10892
\(498\) 0 0
\(499\) 17.6430 0.789808 0.394904 0.918722i \(-0.370778\pi\)
0.394904 + 0.918722i \(0.370778\pi\)
\(500\) 0 0
\(501\) 11.8304 0.528541
\(502\) 0 0
\(503\) 24.6202 1.09776 0.548880 0.835901i \(-0.315054\pi\)
0.548880 + 0.835901i \(0.315054\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 1.41204 0.0625876 0.0312938 0.999510i \(-0.490037\pi\)
0.0312938 + 0.999510i \(0.490037\pi\)
\(510\) 0 0
\(511\) −28.8113 −1.27453
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −35.3516 −1.55476
\(518\) 0 0
\(519\) −11.6634 −0.511968
\(520\) 0 0
\(521\) 38.5298 1.68802 0.844011 0.536326i \(-0.180188\pi\)
0.844011 + 0.536326i \(0.180188\pi\)
\(522\) 0 0
\(523\) −2.23353 −0.0976653 −0.0488326 0.998807i \(-0.515550\pi\)
−0.0488326 + 0.998807i \(0.515550\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −5.21455 −0.226720
\(530\) 0 0
\(531\) −7.53896 −0.327163
\(532\) 0 0
\(533\) 0.198578 0.00860139
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.5403 0.541155
\(538\) 0 0
\(539\) 14.1823 0.610876
\(540\) 0 0
\(541\) −37.8892 −1.62898 −0.814492 0.580175i \(-0.802984\pi\)
−0.814492 + 0.580175i \(0.802984\pi\)
\(542\) 0 0
\(543\) 5.44343 0.233600
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.8645 1.87551 0.937755 0.347299i \(-0.112901\pi\)
0.937755 + 0.347299i \(0.112901\pi\)
\(548\) 0 0
\(549\) 11.9259 0.508985
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −26.3357 −1.11991
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.4537 −0.570050 −0.285025 0.958520i \(-0.592002\pi\)
−0.285025 + 0.958520i \(0.592002\pi\)
\(558\) 0 0
\(559\) −5.70861 −0.241448
\(560\) 0 0
\(561\) 34.2574 1.44635
\(562\) 0 0
\(563\) −31.6140 −1.33237 −0.666186 0.745785i \(-0.732074\pi\)
−0.666186 + 0.745785i \(0.732074\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.09417 −0.129943
\(568\) 0 0
\(569\) −22.3562 −0.937222 −0.468611 0.883405i \(-0.655245\pi\)
−0.468611 + 0.883405i \(0.655245\pi\)
\(570\) 0 0
\(571\) −12.6596 −0.529788 −0.264894 0.964277i \(-0.585337\pi\)
−0.264894 + 0.964277i \(0.585337\pi\)
\(572\) 0 0
\(573\) −5.22887 −0.218439
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.48861 −0.311755 −0.155877 0.987776i \(-0.549820\pi\)
−0.155877 + 0.987776i \(0.549820\pi\)
\(578\) 0 0
\(579\) −17.5287 −0.728469
\(580\) 0 0
\(581\) 45.7707 1.89889
\(582\) 0 0
\(583\) 22.1995 0.919411
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.41858 −0.182374 −0.0911872 0.995834i \(-0.529066\pi\)
−0.0911872 + 0.995834i \(0.529066\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 22.4764 0.924557
\(592\) 0 0
\(593\) 12.5810 0.516641 0.258320 0.966059i \(-0.416831\pi\)
0.258320 + 0.966059i \(0.416831\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.9143 0.692256
\(598\) 0 0
\(599\) −2.64337 −0.108005 −0.0540025 0.998541i \(-0.517198\pi\)
−0.0540025 + 0.998541i \(0.517198\pi\)
\(600\) 0 0
\(601\) −10.4740 −0.427243 −0.213622 0.976916i \(-0.568526\pi\)
−0.213622 + 0.976916i \(0.568526\pi\)
\(602\) 0 0
\(603\) 4.83172 0.196763
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.0197 0.487863 0.243932 0.969792i \(-0.421563\pi\)
0.243932 + 0.969792i \(0.421563\pi\)
\(608\) 0 0
\(609\) −5.28674 −0.214229
\(610\) 0 0
\(611\) −6.41586 −0.259558
\(612\) 0 0
\(613\) 34.7676 1.40425 0.702124 0.712054i \(-0.252235\pi\)
0.702124 + 0.712054i \(0.252235\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.6644 1.27476 0.637380 0.770549i \(-0.280018\pi\)
0.637380 + 0.770549i \(0.280018\pi\)
\(618\) 0 0
\(619\) 22.6024 0.908469 0.454234 0.890882i \(-0.349913\pi\)
0.454234 + 0.890882i \(0.349913\pi\)
\(620\) 0 0
\(621\) −4.21728 −0.169234
\(622\) 0 0
\(623\) 29.1069 1.16614
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.0490 0.998769
\(630\) 0 0
\(631\) 18.6229 0.741366 0.370683 0.928759i \(-0.379124\pi\)
0.370683 + 0.928759i \(0.379124\pi\)
\(632\) 0 0
\(633\) 11.8391 0.470562
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.57391 0.101982
\(638\) 0 0
\(639\) 7.98977 0.316070
\(640\) 0 0
\(641\) −8.77658 −0.346654 −0.173327 0.984864i \(-0.555452\pi\)
−0.173327 + 0.984864i \(0.555452\pi\)
\(642\) 0 0
\(643\) −30.6920 −1.21037 −0.605186 0.796084i \(-0.706901\pi\)
−0.605186 + 0.796084i \(0.706901\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.2023 1.42326 0.711629 0.702555i \(-0.247958\pi\)
0.711629 + 0.702555i \(0.247958\pi\)
\(648\) 0 0
\(649\) −41.5399 −1.63058
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −42.8746 −1.67781 −0.838906 0.544277i \(-0.816804\pi\)
−0.838906 + 0.544277i \(0.816804\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.31145 0.363274
\(658\) 0 0
\(659\) −3.72907 −0.145264 −0.0726320 0.997359i \(-0.523140\pi\)
−0.0726320 + 0.997359i \(0.523140\pi\)
\(660\) 0 0
\(661\) −13.2057 −0.513642 −0.256821 0.966459i \(-0.582675\pi\)
−0.256821 + 0.966459i \(0.582675\pi\)
\(662\) 0 0
\(663\) 6.21728 0.241459
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.20569 −0.279005
\(668\) 0 0
\(669\) −4.48246 −0.173302
\(670\) 0 0
\(671\) 65.7120 2.53678
\(672\) 0 0
\(673\) −0.901611 −0.0347546 −0.0173773 0.999849i \(-0.505532\pi\)
−0.0173773 + 0.999849i \(0.505532\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.0057 −0.384552 −0.192276 0.981341i \(-0.561587\pi\)
−0.192276 + 0.981341i \(0.561587\pi\)
\(678\) 0 0
\(679\) −44.2547 −1.69834
\(680\) 0 0
\(681\) 6.35799 0.243639
\(682\) 0 0
\(683\) −16.9110 −0.647082 −0.323541 0.946214i \(-0.604873\pi\)
−0.323541 + 0.946214i \(0.604873\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.45502 −0.246274
\(688\) 0 0
\(689\) 4.02893 0.153490
\(690\) 0 0
\(691\) 43.1003 1.63961 0.819807 0.572640i \(-0.194081\pi\)
0.819807 + 0.572640i \(0.194081\pi\)
\(692\) 0 0
\(693\) −17.0490 −0.647638
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.23462 −0.0467645
\(698\) 0 0
\(699\) 6.40290 0.242180
\(700\) 0 0
\(701\) 34.9170 1.31880 0.659399 0.751793i \(-0.270811\pi\)
0.659399 + 0.751793i \(0.270811\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.8113 0.782688
\(708\) 0 0
\(709\) 20.8140 0.781685 0.390843 0.920457i \(-0.372184\pi\)
0.390843 + 0.920457i \(0.372184\pi\)
\(710\) 0 0
\(711\) 8.51139 0.319202
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.55793 0.0581821
\(718\) 0 0
\(719\) 10.6024 0.395404 0.197702 0.980262i \(-0.436652\pi\)
0.197702 + 0.980262i \(0.436652\pi\)
\(720\) 0 0
\(721\) 52.8421 1.96794
\(722\) 0 0
\(723\) 14.1911 0.527772
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.06524 −0.0395076 −0.0197538 0.999805i \(-0.506288\pi\)
−0.0197538 + 0.999805i \(0.506288\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 35.4920 1.31272
\(732\) 0 0
\(733\) −29.1216 −1.07563 −0.537816 0.843062i \(-0.680750\pi\)
−0.537816 + 0.843062i \(0.680750\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.6229 0.980667
\(738\) 0 0
\(739\) −41.7966 −1.53751 −0.768757 0.639541i \(-0.779125\pi\)
−0.768757 + 0.639541i \(0.779125\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.8531 1.20526 0.602632 0.798019i \(-0.294119\pi\)
0.602632 + 0.798019i \(0.294119\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −14.7926 −0.541231
\(748\) 0 0
\(749\) −23.6225 −0.863146
\(750\) 0 0
\(751\) −20.2956 −0.740597 −0.370299 0.928913i \(-0.620745\pi\)
−0.370299 + 0.928913i \(0.620745\pi\)
\(752\) 0 0
\(753\) −22.7082 −0.827533
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.62331 −0.0590000 −0.0295000 0.999565i \(-0.509392\pi\)
−0.0295000 + 0.999565i \(0.509392\pi\)
\(758\) 0 0
\(759\) −23.2373 −0.843462
\(760\) 0 0
\(761\) −50.4410 −1.82848 −0.914242 0.405169i \(-0.867213\pi\)
−0.914242 + 0.405169i \(0.867213\pi\)
\(762\) 0 0
\(763\) 15.5329 0.562330
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.53896 −0.272216
\(768\) 0 0
\(769\) −6.75339 −0.243533 −0.121767 0.992559i \(-0.538856\pi\)
−0.121767 + 0.992559i \(0.538856\pi\)
\(770\) 0 0
\(771\) 21.4172 0.771322
\(772\) 0 0
\(773\) 36.5660 1.31519 0.657594 0.753373i \(-0.271574\pi\)
0.657594 + 0.753373i \(0.271574\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −12.4662 −0.447223
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 44.0239 1.57530
\(782\) 0 0
\(783\) 1.70861 0.0610607
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6.21426 0.221514 0.110757 0.993847i \(-0.464672\pi\)
0.110757 + 0.993847i \(0.464672\pi\)
\(788\) 0 0
\(789\) 26.4951 0.943252
\(790\) 0 0
\(791\) −51.4342 −1.82879
\(792\) 0 0
\(793\) 11.9259 0.423501
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.5788 1.40195 0.700977 0.713184i \(-0.252748\pi\)
0.700977 + 0.713184i \(0.252748\pi\)
\(798\) 0 0
\(799\) 39.8892 1.41118
\(800\) 0 0
\(801\) −9.40699 −0.332380
\(802\) 0 0
\(803\) 51.3064 1.81056
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.7259 −0.518378
\(808\) 0 0
\(809\) −3.47236 −0.122082 −0.0610408 0.998135i \(-0.519442\pi\)
−0.0610408 + 0.998135i \(0.519442\pi\)
\(810\) 0 0
\(811\) −29.2404 −1.02677 −0.513384 0.858159i \(-0.671608\pi\)
−0.513384 + 0.858159i \(0.671608\pi\)
\(812\) 0 0
\(813\) 25.8491 0.906566
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −3.09417 −0.108119
\(820\) 0 0
\(821\) 48.3417 1.68714 0.843569 0.537020i \(-0.180450\pi\)
0.843569 + 0.537020i \(0.180450\pi\)
\(822\) 0 0
\(823\) −9.33889 −0.325533 −0.162767 0.986665i \(-0.552042\pi\)
−0.162767 + 0.986665i \(0.552042\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.8175 −1.59323 −0.796616 0.604486i \(-0.793379\pi\)
−0.796616 + 0.604486i \(0.793379\pi\)
\(828\) 0 0
\(829\) −37.8819 −1.31569 −0.657847 0.753151i \(-0.728533\pi\)
−0.657847 + 0.753151i \(0.728533\pi\)
\(830\) 0 0
\(831\) −20.1227 −0.698049
\(832\) 0 0
\(833\) −16.0027 −0.554462
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −32.6678 −1.12782 −0.563909 0.825837i \(-0.690703\pi\)
−0.563909 + 0.825837i \(0.690703\pi\)
\(840\) 0 0
\(841\) −26.0807 −0.899333
\(842\) 0 0
\(843\) 16.3954 0.564687
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −59.9046 −2.05835
\(848\) 0 0
\(849\) 14.5650 0.499871
\(850\) 0 0
\(851\) −16.9911 −0.582448
\(852\) 0 0
\(853\) 26.4867 0.906887 0.453443 0.891285i \(-0.350195\pi\)
0.453443 + 0.891285i \(0.350195\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.1292 1.47327 0.736634 0.676292i \(-0.236414\pi\)
0.736634 + 0.676292i \(0.236414\pi\)
\(858\) 0 0
\(859\) −41.1690 −1.40467 −0.702334 0.711848i \(-0.747859\pi\)
−0.702334 + 0.711848i \(0.747859\pi\)
\(860\) 0 0
\(861\) 0.614436 0.0209399
\(862\) 0 0
\(863\) 7.88822 0.268518 0.134259 0.990946i \(-0.457135\pi\)
0.134259 + 0.990946i \(0.457135\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −21.6546 −0.735428
\(868\) 0 0
\(869\) 46.8980 1.59091
\(870\) 0 0
\(871\) 4.83172 0.163716
\(872\) 0 0
\(873\) 14.3026 0.484069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.03740 −0.203869 −0.101934 0.994791i \(-0.532503\pi\)
−0.101934 + 0.994791i \(0.532503\pi\)
\(878\) 0 0
\(879\) 26.6995 0.900551
\(880\) 0 0
\(881\) −41.4546 −1.39664 −0.698321 0.715785i \(-0.746069\pi\)
−0.698321 + 0.715785i \(0.746069\pi\)
\(882\) 0 0
\(883\) 35.0575 1.17978 0.589889 0.807484i \(-0.299172\pi\)
0.589889 + 0.807484i \(0.299172\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.8661 0.902075 0.451038 0.892505i \(-0.351054\pi\)
0.451038 + 0.892505i \(0.351054\pi\)
\(888\) 0 0
\(889\) 8.90161 0.298550
\(890\) 0 0
\(891\) 5.51003 0.184593
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −4.21728 −0.140811
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −25.0490 −0.834503
\(902\) 0 0
\(903\) −17.6634 −0.587802
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −38.2137 −1.26887 −0.634433 0.772978i \(-0.718766\pi\)
−0.634433 + 0.772978i \(0.718766\pi\)
\(908\) 0 0
\(909\) −6.72595 −0.223086
\(910\) 0 0
\(911\) −49.8919 −1.65299 −0.826496 0.562942i \(-0.809669\pi\)
−0.826496 + 0.562942i \(0.809669\pi\)
\(912\) 0 0
\(913\) −81.5074 −2.69750
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 51.1153 1.68798
\(918\) 0 0
\(919\) 5.41599 0.178657 0.0893284 0.996002i \(-0.471528\pi\)
0.0893284 + 0.996002i \(0.471528\pi\)
\(920\) 0 0
\(921\) −11.4230 −0.376400
\(922\) 0 0
\(923\) 7.98977 0.262986
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −17.0779 −0.560913
\(928\) 0 0
\(929\) −58.7311 −1.92691 −0.963453 0.267878i \(-0.913678\pi\)
−0.963453 + 0.267878i \(0.913678\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −19.4546 −0.636916
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.01541 −0.196515 −0.0982574 0.995161i \(-0.531327\pi\)
−0.0982574 + 0.995161i \(0.531327\pi\)
\(938\) 0 0
\(939\) 32.5805 1.06322
\(940\) 0 0
\(941\) 40.7389 1.32805 0.664025 0.747710i \(-0.268847\pi\)
0.664025 + 0.747710i \(0.268847\pi\)
\(942\) 0 0
\(943\) 0.837461 0.0272715
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.69715 0.120141 0.0600705 0.998194i \(-0.480867\pi\)
0.0600705 + 0.998194i \(0.480867\pi\)
\(948\) 0 0
\(949\) 9.31145 0.302263
\(950\) 0 0
\(951\) 31.3781 1.01750
\(952\) 0 0
\(953\) −1.12460 −0.0364292 −0.0182146 0.999834i \(-0.505798\pi\)
−0.0182146 + 0.999834i \(0.505798\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.41450 0.304327
\(958\) 0 0
\(959\) −39.7614 −1.28396
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 7.63450 0.246018
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.85956 −0.188431 −0.0942153 0.995552i \(-0.530034\pi\)
−0.0942153 + 0.995552i \(0.530034\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.0737 1.06138 0.530692 0.847565i \(-0.321932\pi\)
0.530692 + 0.847565i \(0.321932\pi\)
\(972\) 0 0
\(973\) 52.7526 1.69117
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21.1488 −0.676610 −0.338305 0.941037i \(-0.609853\pi\)
−0.338305 + 0.941037i \(0.609853\pi\)
\(978\) 0 0
\(979\) −51.8328 −1.65658
\(980\) 0 0
\(981\) −5.02006 −0.160278
\(982\) 0 0
\(983\) −32.8960 −1.04922 −0.524610 0.851343i \(-0.675789\pi\)
−0.524610 + 0.851343i \(0.675789\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −19.8518 −0.631889
\(988\) 0 0
\(989\) −24.0748 −0.765534
\(990\) 0 0
\(991\) −18.6322 −0.591871 −0.295935 0.955208i \(-0.595631\pi\)
−0.295935 + 0.955208i \(0.595631\pi\)
\(992\) 0 0
\(993\) 30.4951 0.967734
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.08298 0.0976389 0.0488195 0.998808i \(-0.484454\pi\)
0.0488195 + 0.998808i \(0.484454\pi\)
\(998\) 0 0
\(999\) 4.02893 0.127470
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7800.2.a.bt.1.2 4
5.2 odd 4 1560.2.l.d.1249.7 yes 8
5.3 odd 4 1560.2.l.d.1249.3 8
5.4 even 2 7800.2.a.by.1.3 4
15.2 even 4 4680.2.l.g.2809.3 8
15.8 even 4 4680.2.l.g.2809.4 8
20.3 even 4 3120.2.l.n.1249.7 8
20.7 even 4 3120.2.l.n.1249.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.d.1249.3 8 5.3 odd 4
1560.2.l.d.1249.7 yes 8 5.2 odd 4
3120.2.l.n.1249.3 8 20.7 even 4
3120.2.l.n.1249.7 8 20.3 even 4
4680.2.l.g.2809.3 8 15.2 even 4
4680.2.l.g.2809.4 8 15.8 even 4
7800.2.a.bt.1.2 4 1.1 even 1 trivial
7800.2.a.by.1.3 4 5.4 even 2